Solve the Quadratic Equation by Factorization: 1/(x + 4) − 1/(x − 7) = 11/30

Question:

$$ \frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}, \quad x\ne -4,7 $$

Solution

$$ \frac{(x-7)-(x+4)}{(x+4)(x-7)}=\frac{11}{30} $$

$$ \frac{-11}{(x+4)(x-7)}=\frac{11}{30} $$

$$ (x+4)(x-7)=-30 $$

$$ x^2-3x-28=-30 $$

$$ x^2-3x+2=0 $$

$$ x^2-2x-x+2=0 $$

$$ x(x-2)-1(x-2)=0 $$

$$ (x-2)(x-1)=0 $$

Either

$$ x-2=0 $$

$$ x=2 $$

or

$$ x-1=0 $$

$$ x=1 $$

Hence,

$$ \boxed{x=2,\ 1} $$